Maximally multipartite entangled states

18 downloads 0 Views 140KB Size Report
Jun 12, 2008 - Paolo Facchi,1, 2 Giuseppe Florio,3, 2 Giorgio Parisi,4 and Saverio Pascazio3, 2. 1Dipartimento di Matematica, Universit`a di Bari, I-70125 Bari ...
Maximally multipartite entangled states Paolo Facchi,1, 2 Giuseppe Florio,3, 2 Giorgio Parisi,4 and Saverio Pascazio3, 2 1

arXiv:0710.2868v2 [quant-ph] 12 Jun 2008

Dipartimento di Matematica, Universit` a di Bari, I-70125 Bari, Italy 2 INFN, Sezione di Bari, I-70126 Bari, Italy 3 Dipartimento di Fisica, Universit` a di Bari, I-70126 Bari, Italy 4 Dipartimento di Fisica, Universit` a di Roma ”La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy Centre for Statistical Mechanics and Complexity (SMC), CNR-INFM, 00185 Roma, Italy INFN, Sezione di Roma, 00185 Roma, Italy (Dated: June 12, 2008) We introduce the notion of maximally multipartite entangled states of n qubits as a generalization of the bipartite case. These pure states have a bipartite entanglement that does not depend on the bipartition and is maximal for all possible bipartitions. They are solutions of a minimization problem. Examples for small n are investigated, both analytically and numerically. PACS numbers: 03.67.Mn; 03.65.Ud; 89.75.-k

The characterization of multipartite entanglement is no simple matter. The bipartite entanglement of a composed quantum system [1] can be consistently defined and quantified in terms of the entropy of entanglement or some physically equivalent quantity. On the other hand, there is no unique way of characterizing the multipartite entanglement. Different definitions often do not agree with each other, because they adopt different strategies, focus on different aspects and capture different features of this quantum phenomenon [2, 3, 4]. There is a profound reason behind this problem: the number of measures (i.e. real numbers) needed to quantify multipartite entanglement grows exponentially with the size of the system (e.g., the number of qubits). Therefore, the definition of appropriate entanglement measures, able to summarize the most salient global features of entanglement, can be very difficult and their evaluation bear serious computational difficulties. This difficulty is a characteristic trait of complexity [6] and entanglement is no exception in this respect [7, 8]: we shall introduce here multipartite entangled states that bear the symptoms of frustration. The aim of this Letter is to explore “maximally” multipartite entangled states of n qubits. These states, to be precisely defined later, are maximally (bipartite) entangled for all possible bipartitions. The focus is therefore on the global, partition-independent, features of entanglement. We will consider only pure states, the extension to mixed states being not straightforward, due to wellknown phenomena such as bound entanglement [9]. We consider an ensemble S = {1, 2, . . . , n} of n qubits in the Hilbert space H = (C2 )⊗n , whose state is |ψi =

X

k∈Z2n

zk |ki,

zk ∈ C,

X

k∈Z2n

2

|zk | = 1,

(1)

where k = (ki )i∈S , with ki ∈ Z2 = {0, 1}, and |ki =

O i∈S

|ki i,

|ki i ∈ C2i .

(2)

In order to analyze the multipartite features of the entanglement shared by the qubits, we proceed as follows. ¯ of the system, made up of Consider a bipartition (A, A) nA and nA¯ qubits, respectively, where A ⊂ S is a subset of nA elements, A¯ = S\A its complement, nA + nA¯ = n and we will stipulate nA ≤ nA¯ with no loss of generality. The total Hilbert space isNaccordingly factorized into H = HA ⊗ HA¯ , with HA = i∈A C2i , of dimensions NA = 2nA and NA¯ = 2nA¯ , respectively (NA NA¯ = N ). As a measure of the bipartite entanglement between the two subsets, we consider the purity of subsystem A πA = TrA ρ2A ,

ρA = TrA¯ |ψihψ|,

(3)

TrX being the partial trace over subsystem X. We notice that πA = πA¯ and 1/NA ≤ πA ≤ 1.

(4)

State (1) can be written accordingly to the bipartition ¯ as (A, A) |ψi =

X

k∈Z2n

¯

zk |k A i ⊗ |k A i,

(5)

N where |k A i = i∈A |ki i ∈ HA . By plugging Eq. (5) into Eq. (3) we obtain πA =

X

zk1 z¯k2 zk3 z¯k4 δk1A ,k4A δk2A ,k3A δkA¯ ,kA¯ δkA¯ ,kA¯ . (6) 1

2

3

4

ki ∈Z2n

Notice that for a given bipartition it is very easy to saturate the lower bound 1/NA of (6): one looks for those maximally bipartite entagled states that yield a totally mixed state ρA = 11/NA . We will generalize the above property by requiring maximal possible mixedness for each subsystem A ⊂ S, given the constraint that the total system be in a pure state. A state endowed with this property will be called a maximally multipartite entangled state (MMES).

2 In the most favorable case this means that every subsystem A composed of nA ≤ n/2 qubits is in a totally mixed state ρA = 11/NA and thus πA = 1/NA (recall that πA = πA¯ , thus when nA > n/2 the above require¯ In fact, it is sufficient to look at ment applies to A). maximal subsystems of size nA = [n/2] ([x] =integer part of x), because the density matrix of every smaller part B ⊂ A would automatically be of the sought form, ρB = ρA = 11/NB . Therefore a perfect MMES would TrB∩A ¯ ¯ and be maximally entangled for every bipartition (A, A) would be characterized by πA = 1/NA for all balanced bipartitions. Observe that the requirement of maximal ¯ πA = 1/NA , and the analogous mixedness for (A, A), ¯ , requirement for a different balanced bipartition (B, B) with B 6= A, might not be compatible with each other, so that, at variance with the bipartite case, perfect MMES do not necessarily exist. We define a MMES as a minimizer of what we shall call the potential of multipartite entanglement −1 X  n πA , (7) πME = nA |A|=nA

where nA = [n/2]. The above quantity is related to the A (average) linear entropy SL = NN (1−πME ) introduced A −1 in [4], that extends ideas put forward in [3]. See also [5]. The quantity πME measures the average bipartite entanglement over all possible balanced bipartition and thus inherits property (4), i.e. 1/NA ≤ πME ≤ 1.

(8)

The upper bound πME Q = 1 is attained by the fully factorized states, zk = i∈S αiki , with |αi0 |2 + |αi1 |2 = 1. On the other hand, the lower bound πME = 1/NA , if attained, would correspond to a perfect MMES, maximally entangled for every bipartition. However, it can happen that the requirements of maximal mixedness for different bipartitions compete with each other. In such a case, the system is frustrated and the minimum of the potential (7) is strictly larger than the lower bound in (8), i.e. min πME > 1/NA . Since in such situation it may happen that different bipartitions yield different values of πA , our strategy will be to seek those states among the minimizers that have the smallest variance. This quest can be recast as an optimization problem: search for the minimum of the cost function π ˜ME (λ) = πME + λσME , where λ ≥ 0 is a Lagrange multiplier and −1 X  n 2 2 (πA − πME ) σME = nA

a compromise between the minimal purity πME (maximal average entanglement) and the minimal standard deviation σME of the distribution (maximally distributed entanglement). In general, the solution of this optimization problem completely defines a class of states with the maximal possible entanglement (minimum purity), that is also well distributed, being as insensitive as possible to the particular choice of the bipartition. If λ ≫ 1 the minimization process will yield a very well peaked distribution of πA around its average: entanglement will be uniformly distributed, but this does not necessarily provide a MMES; for example, a completely separable state has a (vanishing) entanglement that is insensitive to the change of the bipartition (πA = 1 for all bipartitions). More interesting is the case λ = 0; indeed, a solution that minimizes the cost function π ˜ME (0) will have an entanglement distribution centered on the minimum of the potential πME . Therefore, if this minimum saturates the lower bound in (8), the width σME must vanish. This would be our desideratum. However, it is known that, for n ≥ 8, perfect MMES do not exist [4]. The general problem is therefore complicated. As a first step we set λ = 0 in Eq. (9), and focus on the minimization of the potential πME ≡ π ˜ME (0). We shall tackle this problem both analytically and numerically. By plugging (6) into (7) one gets after some combinatorics X (11) ∆(k1 , k2 ; k3 , k4 )zk1 zk2 z¯k3 z¯k4 , πME = ki ∈Z2n

with ∆(k1 , k2 ; l1 , l2 ) =

 −1X n δk1A ,lA δ A A δ A¯ A¯ δ A¯ A¯ 1 k2 ,l2 k1 ,l2 k2 ,l1 nA |A|=nA

= g(k1 ⊕ l1 ∨ k2 ⊕ l2 , k1 ⊕ l2 ∨ k2 ⊕ l1 ),  −1   n − |a| − |b| n , g(a, b) = δa∧b,0 nA − |a| nA

(12)

where a⊕b = (ai +bi mod 2)i∈S is the XOR operation, a∨ b = (ai + bi − ai bi )i∈S the OR operation, a ∧ b = (ai bi )i∈S P the AND operation, and |a| = i∈S ai . Equations (11)(12) yield a closed expression for the average purity, that is amenable to analytic and numerical investigation. In order to further simplify the problem, in the following discussion we√will replace rk = |zk | in Eq. (1) with its mean value 1/ N , and focus on the states 1 X iϕk e |ki. |ψi = √ N k∈Z n

(9)

(13)

2

Plugging Eq. (13) into Eq. (6) we find (10)

|A|=nA

is the variance of πA over all balanced bipartitions. Notice that the introduction of λ enables one to look for

NA + NA¯ − 1 N X 2 + 2 cos (ϕplm − ϕpl′ m + ϕpl′ m′ − ϕplm′ ) ,(14) N ′ ′ πA =

l6=l ,m6=m

3 where ϕplm = ϕp−1 (l,m) , p being a permutation such that A = {p(1), p(2), . . . , p(nA )}. This is an interesting formula, that P is worth discussing in detail: i) first of all, if ϕk = i∈S ϕiki in (13), one obtains separable states, that yield the maximum possible value πA = 1 for all bipartitions: indeed the N (NA − 1)(NA¯ − 1)/4 cosines in the summation in (14) are all 1; ii) the first addendum in the right-hand side corresponds to the average entanglement of typical states [8, 10]. Thus, the combination of phases in the summation can increase or reduce the value of the purity with respect to the typical one (at fixed bipartition); iii) in order to get a lower value of purity, one should look for combinations of angles that tend to yield negative cosines. On the other hand, it is also clear that for n > 2 not all cosines can be −1, as this would yield a ¯ purity smaller than 1/NA ; iv) at fixed bipartition (A, A) it is always possible to find combinations of cosines that saturate the lower bound πA = 1/NA . However, when plugged into (11), the requirement that this lower bound be saturated for every A might not be satisfiable. This problem is in general n-dependent. We now explicitly look at the simplest examples. For two qubits we only have one bipartition and the potential of multipartite entanglement reduces to (2)

πME =

3 1 + cos (ϕ0 − ϕ1 − ϕ2 + ϕ3 ) , 4 4

(15)

where the indices of the phases are expressed again in terms of k, in decimal notation. The minimization of (2) the potential consists in solving the equation πME = 1/2 (remember that NA = NA¯ = 2). It is straightforward to obtain ϕ0 − ϕ1 − ϕ2 + ϕ3 = π, which yields the MMES  1  iϕ0 e |0i + eiϕ1 |1i + eiϕ2 |2i − ei(−ϕ0 +ϕ1 +ϕ2 ) |3i . 2 (16) In this degenerate case, multipartite entanglement coincides with bipartite entanglement, and this state is obviously equivalent, up to local operations, to a Bell state. For three qubits one must look for the solutions of (3) πME = 1/2, where

|ψ2 i =

(3)

πME =

5 1 X cos(ϕp(0) + ϕp(7) − ϕp(1) − ϕp(6) ) + 8 48 p + cos(ϕp(2) + ϕp(5) − ϕp(4) − ϕp(3) ) + 2 cos(ϕp(0) + ϕp(3) − ϕp(1) − ϕp(2) )

 + 2 cos(ϕp(7) + ϕp(4) − ϕp(6) − ϕp(5) ) , (17)

where the sum is over the 3 cyclic permutations (of the bits). A class of solutions is

1 iϕ0 (e |0i + eiϕ1 |1i + eiϕ2 |2i − ei(−ϕ0 +ϕ1 +ϕ2 ) |3i 8 + eiϕ4 |4i − ei(−ϕ0 +ϕ1 +ϕ4 ) |5i + e−iϕ6 |6i + ei(−ϕ0 +ϕ1 +ϕ6 ) |7i) (18)

|ψ3 i =

and form a 5-dimensional submanifold. This state includes the GHZ state [11] as a particular case and shares the same properties of the GHZ state for what concerns (3) concurrence and one-tangle. Note that πME contains 12 cosines with different arguments, 6 of which are counted twice. The solution proposed corresponds to 2 cosines = 1, 4 cosines = −β, 4 × 2 cosines = −1 and 2 × 2 cosines = β, with β = cos(ϕ0 − ϕ2 − ϕ4 + ϕ6 ), that sum up to 2 − 4β − 8 + 4β = −6. In fact, there are 3 families of solutions of the form (13), corresponding to the 3 submanifolds Mp = {ϕi |ϕp(0) + ϕp(7) − ϕp(1) − ϕp(6) = 0, ϕp(2) + ϕp(5) − ϕp(4) − ϕp(3) = 0, ϕp(0) + ϕp(3) − ϕp(1) − ϕp(2) = π} with p a cyclic permutation. All three classes yield the (3) same pattern of cosines in πME (β being given by the corresponding permutation). For a number of qubits larger than 3, we turned to a numerical approach: we generated a typical state of the form (13) and numerically tackled the minimization problem through different kinds of iterative algorithms (for a review of numerical techniques and their implementation, see [12]). We first used deterministic algorithms. In general, we found that minimization is strongly dependent on the initial conditions. Therefore, we sampled a large number of initial states, in order to test the reliability of the solutions obtained. Among others, the truncated Newton method gave us the best results in terms of both reliability and speed. The use of stochastic algorithms gave us comparable results. In both cases the existence of a large number of degenerate (local and global) minima required an accurate analysis. (4) For n = 4 qubits, we numerically obtained min πME ≃ −4 0.333 √ > 1/4 with σME ≃ 10 . If the requirement |zk | = 1/ N in Eq. (1) [and (13)] is relaxed, one can make σME vanish. This is a first example of frustration among the bipartitions, that prevents the existence of a perfect MMES. It is curious that the requirement that purity be minimal for all balanced bipartitions generate conflicts already for n = 4 qubits. This is consistent with results obtained by other authors [13, 14]. For n = 5 and 6, the landscape of the manifold on which the minimization is performed becomes complicated. Nonetheless, we found perfect MMES, namely (5) (6) solutions for πME = 1/4 and πME = 1/8, respectively. Therefore, and curiously, frustration is present for n = 4 qubits, while it is absent for n = 5 and 6. For example, a 5-qubits perfect MMES is defined by Eq. (13) with the following set of phases (ϕk ) = (0, 0, 0, 0, 0, π, π, 0, 0, π, π, 0, 0, 0, 0, 0, 0, 0, π, π, 0, π, 0, π, π, 0, π, 0, π, π, 0, 0)

(19)

and lives on a 7-dimensional manifold. The distribution of the angles x = ϕpjl −ϕpjl′ +ϕpj′ l′ −ϕpjl′ for a given 6-qubit MMES is displayed in Figure 1. We observe interesting features, shared by all the MMES we investigated and for all values of n: first of all, the distribution is sym-

4 ð of phases 14000

TABLE I: Perfect MMES for different n. n perfect MMES 2,3,5,6 exist 4 do not exist 7 ? ≥8 do not exist

10000 6000 2000 А2

Π

3А2



x

(6) πME ,

FIG. 1: Distribution of the arguments x of the cosines in for a given 6-qubit MMES. Notice the symmetries around π and π/2. The total number of phases is 62720. Different phases are fully resolved by the binning.

metric around x = π; second, there is a large number of instances such that cos x = −1, partially compensated by the contribution of cos x = 1; third, the distribution of the remaining angles is symmetric around x = π/2 and yields a vanishing contribution to πME . For n = 7 we numerically found configurations with (7) min πME ≃ 0.134 > 1/8 and σME ≃ 10−2 , which improves previous bounds [4, 14]. By minimizing the cost function (9) with a non vanishing λ, one can reduce σME to ≃ 10−3 , at the expense of a higher πME ≃ 0.136. It is not clear at present if the impossibility to reach the absolute minimum for n = 7 is to be ascribed to the numerical procedure. For n = 8 and 9, where perfect MMES do not exist [4], the convergence of the numerical simulations become very slow. This is a typical signature of frustration. The numerical results show that, in order to obtain a vanishing width of the distribution for frustrated systems, it is necessary to increase the value of the average of the purity. These conclusions are summarized in Table I. A comment is now in order. Although in this article perfect MMES are exhibited as minimizers of the potential of entanglement, they are, by their very nature, independent of the method used to find them. In fact, by virtue of their maximal mixedness, they saturate all measures of multipartite entanglement. This is the case, for example, of the global entanglement measure of Meyer and Wallach [3], as well as its generalizations. Indeed, any entanglement monotones, being functions of the partial density matrices of subsystems of qubits, attain their maximal values on perfect MMES. The minimization procedure we propose is just a convenient way to construct them. For non-perfect MMES, due to a possible finite value of σME , different measures of entanglement can resolve a part of the degeneracy of the manifold of minimizers. It is interesting to briefly discuss one straightforward potential application. Consider the 5-qubit perfect MMES |ψ5 i defined by Eqs. (13) and (19). One can easily prove that hψ5 |σ1z σ2z σ3y σ4y σ5z |ψ5 i = 1,

(20)

where σiy and σiz are Pauli matrices. Therefore, the single- and two-qubit statistics are always “flat”, but the measurements of the observables in (20) are always strictly correlated. Any two parties, that can be far apart, can therefore share a cryptographic key only if the other three parties agree on measuring their respective observables in (20) and making their results public. Notice that the key is shared by the two parties but is unknown to the other three. We call this phenomenon majority-agreed key distribution. In conclusion, we introduced a class of multipartite entangled states that maximize the amount and distribution of entanglement. The features of these states strongly depend on the number of qubits involved. In our numerical search, we noticed that already for a relatively small number of qubits (n ≥ 7), the landscape of the parameter space where the optimization procedure is performed has a complex structure with a large number of local minima and a very slow convergence. The presence of frustration, due to the competition among different partitions (observed already for n = 4), appears to be a general feature of many-body systems. It prevents the possibility to find perfect MMES but introduces interesting perspectives. In this sense the minimization task is a problem that requires a careful analysis and the use of numerical and analytical strategies from different research fields. In this paper we just started to explore these connections. Thus, the study of maximally multipartite entangled states paves the way towards a deeper comprehension of the complex structure of quantum correlations arising in many-body systems. This work is partly supported by the European Community through the Integrated Project EuroSQIP.

[1] W. K. Wootters, Quantum Inf. and Comp., 1, 27 (2001); M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007); L. Amico, R. Fazio, A. Osterloh and V. Vedral “Entanglement in Many-Body Systems,” arXiv:quant-ph/0703044 (Rev. Mod. Phys., in print). [2] V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, 052306 (2000); A. Wong and N. Christensen, Phys. Rev. A 63, 044301 (2001); D. Bruss, J. Math. Phys. 43, 4237 (2002); G. Rigolin, T. R. de Oliveira and M. C. de Oliveira, Phys. Rev. A 74, 022314 (2006). [3] D.A. Meyer and N.R. Wallach, J. Math. Phys. 43, 4273 (2002). [4] A. J. Scott, Phys. Rev. A 69, 052330 (2004).

5 [5] K. R. Parthasarathy, Proc. Indian Acad. Sciences, 114, 365 (2004). [6] M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987). [7] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, J. Phys. A: Math. Gen. 35, 7137 (2002). [8] P. Facchi, G. Florio and S. Pascazio, Phys. Rev. A 74, 042331 (2006); Int. J. Quantum Inf. 5, 97 (2007). [9] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998); C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999). [10] E. Lubkin, J. Math. Phys. 19, 1028 (1978); S. Lloyd and H. Pagels, Ann. Phys., NY, 188, 186 (1988); D. N. Page, ˙ Phys. Rev. Lett. 71, 1291 (1993); K. Zyczkowski and H.-

[11] [12] [13]

[14]

J. Sommers, J. Phys. A 34, 7111 (2001); A. J. Scott and C. M. Caves, J. Phys. A: Math. Gen. 36, 9553 (2003). D.M. Greenberger, M. Horne and A. Zeilinger, Am. J. Phys. 58, 1131 (1990). W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran 77, Numerical Recipes in C (Cambridge University Press, 1992). A. Higuchi and A. Sudbery, Phys. Lett. A273, 213 (2000); S. Brierley and A. Higuchi, J. Phys. A: Math. Theor. 40, 8455 (2007). A. Borras, A.R. Plastino, J. Batle, C. Zander, M. Casas, and A. Plastino, J. Phys. A: Math. Theor. 40, 13407 (2007).